Soluciones simétricas de algunos problemas elípticos

In this paper we study solutions to the Neumann problem (I)         ∆u=  F(u)   in Ω,    ∂u/∂n =  G(u)  on Ω,                                                and the Dirichlet problema      (II)    ∆u=F(u)   in  Ω,               u=c        n  ∂Ω      where Ω is a bounded domain in Rn with a smooth boundary ∂ Ω  ∂/ ∂n is the derivative with respect to the outward normal n and c ϵ R. If  Ω is the unit ball and if either F(t) = f(t) and G(t) = g(t) or F(t) = /(t) . t and G(t) = 9(t) . t where f is a strictly increasing continuous function and  g is a strictly decreasing continuous function, we prove that solutions to problems (I) and (II) are radially symmetric about the origen. If Ω  is the unit ball and F is a continuous function that does not change sign, we prove that solutions of (II) are radially symmetric about the origen. If Ω ⊂ Rn  is a symmetric bounded domain with respect to a hyperplane T and f ϵ C(Ω x R,R), g ϵC (∂Ω x R, R) are functions that satisfy the same monoton..icity properties in the second variable as before, then we prove that solutions are symmetric with respect to the hyperplane T. If F satisfies the same condition as in the first case and G ≡ 0, we prove that the only solutions of (I) are constant functions. Furthermore, we find a formula for solutions of (I) in the unitary ball that allow us to deduce some non-existence results. We find conditions on F and G in order for (I) to have no solutions in any bounded domain.

• Autores: Quintero H., José Raúl
• Tipo de recurso: Article of journal
• Fecha de publicación: 1993
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: spa
• OAI Identifier: oai:repositorio.unal.edu.co:unal/43591
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/43591; http://bdigital.unal.edu.co/33689/
• Palabra clave: Bounded domain; soft limit; derivative; continuous function; hyperplane
• Rights: openAccess
• License: Atribución-NoComercial 4.0 Internacional